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Full text of "The Wealth and Well-Being Curves, and Political Freedom: A Mathematical Analysis: Part 1"

The Wealth and Weil-Being Curves, and Political 
Freedom - A Mathematical Analysis: Part 1 



By Patrick Bruskiewich 
Mathematisches Institut der Vancouver, BC 



Abstract 

In a recent newspaper the author came across a scatter plot of Wealth (GDP) versus Well- 
Being. The distribution is similar to those found in Learning Theory and in a number of 
Applied Mathematics models. The scatter plot begs a mathematical analysis as well as a 
correlation to socio-economic measures. 



1.0 The Wealth and the Weil-Being Curve 

In a recent edition of a Canadian newspaper The National Post the author came across an 
article about the puzzling angst of the People of France. With this March 26, 2013 article 
was included a distribution that plotted Gross Domestic Product as an independent 
variable with an implied correlation to a dependent "subjective Well-Being Index" (refer 
to Fig. 1 : GDP versus Well-Being Index). 

The distribution shows a number of indicators that relate or correlate to the "subjective 
Well-Being Index", and representative economic systems. The scatter plot came 
complete with a curve (denoted red in the article) with no description of the functional 
form the fit. The distribution was sourced from the Paris School of Economics. 



A plethora of curves can be fitting to this distribution depending on the variables used. 
Perhaps, to a first approximation a restricted growth algorithm, or Mitscherlich 
relationship, could be used to describe the distribution. [1] 



■ LATIN AMERICA o FORMER COMMUNIST 



4.5 



-LB 
-1.5 
-Z.B 
-2.5 



Puerto Ricn 



o Ireland 



New 

Zealand 



Denmark 
Iceland .>-. 
A Switzerland 

Netherlands • 

Sweden -a * united States^ 

Australia -o J^ mximbowg^ 
Britain BeJoium 
Finland -o 6 A, o- Austria N°™ay-o 




Bangladesh lran 

h 

India 
c 

„ !■ /y o Serbia 
BurkinaXy' 

cardan 

Tanzania 
^Azerbaijan ^^ 

Rwanda o- Estonia 

Latvia-Q o- Lithuania 

Georgia - , jv Russia 

j. Bulgaria -do 

Alhania-o Romania 

J Belarus^ ^ Jkraine 

lraq-©„ O-MoEdcva 
O- Armenia 

9 
ZSmbatae 



Curve with article 
Fitted SWI Curve 



~i ; 1 i 

5 10 15 20 

GDP PER CAPITA, IN THOUSANDS OF DOLLARS 



25 



3D 



SOUH.CK: I'AKIS SCHOOL ("Jl-' ECONOMICS 



JONATI [ON IlIVAiT / NATIONAL POST 



Fig. 1: GDP versus Weil-Being Index 



2.0 Restrictive Growth and the Mitscherlich Relationship 

The Mitscherlich Relationship describes a system governed by the differential equation 

dw 



dt 



= k(B-w) 



where k is a positive constant which determines how fast the differential tends to zero, 
and where B represents an upper bound. There exists an upper bound B for the 
productivity of the system (the line y = B is the asymptote). 

No crop, industry or population can grow or flourish indefinitely. There are limitations 
set by shortages of raw materials, productivity of land and by some other sort of control 
mechanism, such as limits to demand, inherent economic efficiencies and the manner in 
which decisions are made or populations are governed. 

For example, there is a relationship between fertilizer use and crop yield. To first 
approximation, the function y can be considered as a monotonically increasing function. 
If x where the amount of fertilizer applied to the cultivation of a crop, the yield y cannot 
be raised indefinitely by the application of more and more fertilizer. All things being 
equal, the land would, with time, become limited in its crop productivity as the soil 
becomes saturated with fertilizer. 

For such a system a workable approximation to the output or productivity of the land is 
given by the Mitscherlich Relationship, 



y = B(l-e- b ) 



In analyzing the locus of points that is graphed with the original article, the author was 
able to fit a simple Mitscherlich Relationship to the red curve which has the form 



SWI = 7r(\-e kw ) 



l 



where k = — , which appears to be a defining feature on the left hand side of the 
distribution and appears to be an inflection point for the distribution. 



The plot of the Subjective Well-Being Index (SWI) function is the blue line in Fig. 1. 
The graph and the SWI fit (blue curve) begs an in-depth mathematical analysis, as well as 
a correlation on socio-economic indicators. 



3.0 Political Freedom as a Significant Socio-Economic Measure 

Based on a careful reading of history, it is evident that there is a continuum of Political 
Freedom that spans from Totalitarianism at one extreme, to a full and open Liberal 
Democratic Institutions and the Ecole Suisse at the other extreme. An index that runs 
from to 3 can be applied to the Continuum of Political Freedom (refer to Table 1 : 
Continuum of Political Freedom) 






0.5 


1.0 


1.5 


2.0 


2.5 


3.0 


Totalitarian 


Neo- 
Communism 


Autocratic 


Oligarchy 


Capitalism 


Liberal 
Democracy 


Ecole 
Suisse 



Table 1: Continuum of Political Freedom 



To more fully represent a correlation between Wealth and Well-Being, while applying a 
measure of the Continuum of Political Freedom (0 < m < 3), we must look beyond the 
simplicity of the Mitscherlich Relationship. 



4.0 Modeling SWI using the Logistic Equation 

It would seem reasonable to develop a model that is independent of monetary measures 
that depend currency exchange are are subject to inflation and other competitive factors. 

Beyond the Mitscherlich Relationship, restrictive growth can be described by a 
differential equation of the expanded form 

where X is a positive coupling constant . The solution for this expanded expression is 

B 

y ~\ + ke- m 

where k has a positive value. For the case k = 1 and e ~ lBt « 1 we get y « B 1 1 - e~ xm j . 
Based on the SWI functional fit (above) 



10 IOtt 

Using a scaling factor, let 

t = l0m 

Then the coupling factor becomes 

7t 



5.0 Subjective Well-Being and Political Freedom 



Consider a logistic function of the more complete form 



dy 
dm 



= A'(A-y)(B-y) 



where m is the Continuum of Political Freedom ( < m < 3 ) outlined above. The 
solution for this more general expression is the logistic equation 



y = A + 



(B-A) 



l + ke 



A'(B-A)m 



It is worth noting that n appears as the factor B in the Mitscherlich Relationship. Our 
more developed model should involve universal measures such a n . We can do this 
by choosing a function of the form for the Subjective Well-Being Index, 



SWI = -2.0 + 



2n 



\ + ke 



-2.0m 



Such a function should describe a span from a well-being index of -1.5 to 4.25 from 
whence we conjecture that the factor k is of the form 



k = 7u { *- x) * 11.606 



The inflection point is at 



d 



r ^,\ 



dm 



dy ]_ d ro , 
V dm ) dm 



\A'(A-y)(B-y)] = 



which occurs at the value 



y = \(B + A) 



which yields 



y = (7U-2) = \. 14159 



For SWI as a function of Political Freedom ( < m < 3 ) we find the following (refer to 
Table 2: SWI as a function of the Political Freedom m): 



Continuum Measure 
m 


Political Freedom 


Subjective Well- 
Being Index 


Efficiency 





Totalitarian 


-1.5 


-0.37 


0.5 


Neo- Communism 


-0.81 


-0.20 


1.0 


Autocratic 


0.44 


0.11 


1.5 


Oligarchy 


1.98 


0.48 


2.0 


Capitalism 


3.18 


0.78 


2.5 


Liberal Democracy 


3.82 


0.93 


3.0 


Ecole Suisse 


4.10 


1.0 



Table 2: SWI as a function of the Political Freedom Measure m 



A graph of the Subjective Well-Being Index (SWI) as a function of the Political Freedom 
Measure m is given in Fig. 2: The SWI as a function of the Political Freedom Measure m 
with some representative countries plotted on the distribution. 



It is clearly evident the more open forms of government not only provide for a better 
sense of Well-Being and Quality of Life they also, by the nature of the Logistic equation, 
provide for greater opportunity for growth and for economic efficiency. 



SWI as a function of m 



-' 



Denmark 



Suisse 



Ideal 



Brazil 



^* 



-*S nl j, * USA 



Taiwan 



_*_ •» "OTC 

Singapore wf 

Tig Spain W Japan 



China* Poland 




^J" South Korea 



South Africa 
♦ India W Hungary 



5/ 

Russia 



' : 



■JJ 1 Ukraine 
•ff North Korea 



m (Political Frasflom) 



35 1 



Fig. 2: The SWI as a function of the Political Freedom Measure m 

A measure of efficiency is the ratio of the SWI for a Political Freedom Measure m 
compared to the Ideal Measure at m = 4.0 (refer to Fig. 3: Economic Efficiency as a 
Function of the Political Freedom Measure m). 

Clearly the Totalitarian and the neo-Communism Regime are less than efficient in their 
provisioning of goods and services compared to the Open Capitalism, the countries with 
Parliamentary and Liberal Democratic ideals, and the Ecole Suisse. 

6.0 Learning, Training and Development 



In a real sense the SWI function describes the base and the enlightened extremes of 
human behaviour. One could speculate that the Continuum Measure m is a measure of 
Informed -Consent, and a measure s which represents an Efficiency of Governance. 



Bf iciency as a f Lncii on of m 



120 
1.00 
30 
60 



fc 040 



020 



^ 000 



-3 20 



-: io 



-0 60 



m {Ported fT««dom|i 





$ulm 




IJmI I 


CW, ^*^^"*1HA 








s 




/ ^L South *V#» 




> 




1 1 aiy^ 


1 1,5 2 2 5 J 


5 5 




: 




*m»» 








WeithKfrru 





Fig. 3: Economic Efficiency as a Function of the Political Freedom Measure m 

For instance, for a measure of 1 < / < 3 and 1 < C < 3 , a simple model for m would be 

9 V ; 

and where < s < 1 . The form of the SWI Function is similar to mathematical functions 
that describe Learning, Training and Development. This may be touched upon in a 
subsequent paper. 

You will note the anomalous nature of Communist North Korea under Kim and Neo- 
Communist Russia under Putin, compared to for instance the People's Republic of China. 

References: 



[1] E. Batschelet, Introduction to Mathematics for Life Scientists, Springer- Verlag,